Boundary Rényi oscillations in the open XX chain

For the ground state of the semi-infinite XX chain with Fermi momentum $k_F$, we derive the extracted $2k_F$ contribution to the Rényi entropy of its first $\ell$ sites at fixed $α>1$. Its phase $\sin[(2\ell+1)k_F]$ matches the Friedel oscillation at the entangling bond and becomes the even--odd alt